Why singleton set is not open in Q?

Answer 1

In the context of the rational numbers ( \mathbb{Q} ) with the standard topology induced by the real numbers ( \mathbb{R} ), a singleton set ( {q} ) (where ( q ) is a rational number) is not open because for any point ( q ) in ( \mathbb{Q} ), every open interval around ( q ) contains both rational and Irrational Numbers. Therefore, any interval ( (q - \epsilon, q + \epsilon) ) intersects with points outside the singleton set, meaning it cannot be entirely contained within ( {q} ). Thus, singleton sets do not satisfy the definition of an open set in ( \mathbb{Q} ).